113,087 research outputs found
Graded Betti numbers of path ideals of cycles and lines
We use purely combinatorial arguments to give a formula to compute all graded
Betti numbers of path ideals of line graphs and cycles. As a consequence we can
give new and short proofs for the known formulas of regularity and projective
dimensions of path ideals of line graphs
!-Graphs with Trivial Overlap are Context-Free
String diagrams are a powerful tool for reasoning about composite structures
in symmetric monoidal categories. By representing string diagrams as graphs,
equational reasoning can be done automatically by double-pushout rewriting.
!-graphs give us the means of expressing and proving properties about whole
families of these graphs simultaneously. While !-graphs provide elegant proofs
of surprisingly powerful theorems, little is known about the formal properties
of the graph languages they define. This paper takes the first step in
characterising these languages by showing that an important subclass of
!-graphs--those whose repeated structures only overlap trivially--can be
encoded using a (context-free) vertex replacement grammar.Comment: In Proceedings GaM 2015, arXiv:1504.0244
Production matrices for geometric graphs
We present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot numbers, and Fibonacci numbers arise in a natural way, and also new formulas are obtained, such as a formula for the number of non-crossing geometric graphs with root vertex of given degree. The characteristic polynomials of some of these production matrices are also presented. The proofs make use of generating trees and Riordan arrays.Postprint (updated version
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